On the closedness of the linear image of a set with applications to generalized spline functions †

In many problems of approximation theory, especially in the field of spline functions and their generalizations, one seeks a u minimizing ||Tu-f|| for u in some constraint set U; it is usually easy to prove existence of such a u if IT/is closed. We consider hypotheses that will guarantee the closedness of TU, and, more generally, the closedness of the sum of U and a closed linear subspace